Ratio of the perimeter of Bernoulli's lemniscate to its diameter
In mathematics, the lemniscate constantϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of π for the circle.[1] Equivalently, the perimeter of the lemniscate is 2ϖ. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.[2] It also appears in evaluation of the gamma and beta function at certain rational values. The symbol ϖ is a cursive variant of π; see Pi § Variant pi.
Sometimes the quantities 2ϖ or ϖ/2 are referred to as the lemniscate constant.[3][4]
As of 2024 over 1.2 trillion digits of this constant have been calculated.[5]
History
Gauss's constant, denoted by G, is equal to ϖ /π ≈ 0.8346268[6] and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as .[7] By 1799, Gauss had two proofs of the theorem that where is the lemniscate constant.[8]
John Todd named two more lemniscate constants, the first lemniscate constantA = ϖ/2 ≈ 1.3110287771 and the second lemniscate constantB = π/(2ϖ) ≈ 0.5990701173.[9][10][11]
The lemniscate constant and Todd's first lemniscate constant were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd's second lemniscate constant and Gauss's constant were proven transcendental by Theodor Schneider in 1941.[9][12][13] In 1975, Gregory Chudnovsky proved that the set is algebraically independent over , which implies that and are algebraically independent as well.[14][15] But the set (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over .[16] In 1996, Yuri Nesterenko proved that the set is algebraically independent over .[17]
Forms
Usually, is defined by the first equality below, but it has many equivalent forms:[18]
Analogously to the Leibniz formula for π,we have[21][22][23][24][25]where is the L-function of the elliptic curve over ; this means that is the multiplicative function given byand bywhere is the number of solutions of the congruencein variables that are non-negative integers.
Equivalently, is given bywhere such that and is the eta function.[26]
The above result can be equivalently written as(the number is the conductor of ) and also tells us that the BSD conjecture is true for the above .[27]
As a special value of other functions
Let be the minimal weight level new form. Then[28]The -coefficient of is the Ramanujan tau function.
The Machin formula for π is and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula . Analogous formulas can be developed for ϖ, including the following found by Gauss: , where is the lemniscate arcsine.[33]
The lemniscate constant can be rapidly computed by the series[34][35]
A (generalized) continued fraction for π isAn analogous formula for ϖ is[10]
Define Brouncker's continued fraction by[40]Let except for the first equality where . Then[41][42]For example,
In fact, the values of and , coupled with the functional equationdetermine the values of for all .
Simple continued fractions
Simple continued fractions for the lemniscate constant and related constants include[43][44]
Integrals
The lemniscate constant ϖ is related to the area under the curve . Defining , twice the area in the positive quadrant under the curve is In the quartic case,
^ In particular, Siegel proved that if and with are algebraic, then or is transcendental. Here, and are Eisenstein series. The fact that is transcendental follows from and Apostol, T. M. (1990). Modular Functions and Dirichlet Series in Number Theory (Second ed.). Springer. p. 12. ISBN 0-387-97127-0. Siegel, C. L. (1932). "Über die Perioden elliptischer Funktionen". Journal für die reine und angewandte Mathematik (in German). 167: 62–69.
^In particular, Schneider proved that the beta function is transcendental for all such that . The fact that is transcendental follows from and similarly for B and G from Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale". Journal für die reine und angewandte Mathematik. 183 (19): 110–128. doi:10.1515/crll.1941.183.110. S2CID 118624331.
^G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
^G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
^In fact, Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 45
^Nesterenko, Y. V.; Philippon, P. (2001). Introduction to Algebraic Independence Theory. Springer. p. 27. ISBN 3-540-41496-7.
^See:
Cox 1984, p. 281
Finch 2003, pp. 420–422
Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF). In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290.
^Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Springer-Verlag. p. 382–406. ISBN 978-3-642-08142-2.
^"Elliptic curve with LMFDB label 32.a3 (Cremona label 32a2)". The L-functions and modular forms database.
^The function is the unique weight level new form and it satisfies the functional equation
^Rubin, Karl (1987). "Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication". Inventiones mathematicae. 89: 528.
^"Newform orbit 1.12.a.a". The L-functions and modular forms database.
^Levin (2006)
^Hyde (2014) proves the validity of a more general Wallis-like formula for clover curves; here the special case of the lemniscate is slightly transformed, for clarity.
^Hyde, Trevor (2014). "A Wallis product on clovers" (PDF). The American Mathematical Monthly. 121 (3): 237–243. doi:10.4169/amer.math.monthly.121.03.237. S2CID 34819500.
^Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1. p. 140 (eq. 3.34), p. 153. There's an error on p. 153: should be .
^Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1. p. 146, 155
^Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
^Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474.
^"A068467 - Oeis".
^ a bCox 1984, p. 313.
^Levien (2008)
^Cox 1984, p. 312.
^Adlaj, Semjon (2012). "An Eloquent Formula for the Perimeter of an Ellipse" (PDF). American Mathematical Society. p. 1097. One might also observe that the length of the "sine" curve over half a period, that is, the length of the graph of the function sin(t) from the point where t = 0 to the point where t = π , is . In this paper and .
Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Finch, Steven R. (18 August 2003). Mathematical Constants. Cambridge University Press. pp. 420–422. ISBN 978-0-521-81805-6.
External links
"Gauss's constant and where it occurs". www.johndcook.com. 2021-10-17.